Solve -4(x-7) = 5(-2x-4): Step-by-Step Solution

Hey everyone! Today, we're diving into a fun little math problem. We've got the equation -4(x-7) = 5(-2x-4), and our mission is to solve for x. Don't worry; it's not as scary as it looks. We'll break it down step by step so you can follow along easily. Whether you're a student tackling algebra or just someone who enjoys a good math puzzle, this guide is for you. Let's get started and unravel this equation together!

Understanding the Basics of Algebraic Equations

Before we jump into solving our specific equation, let's quickly review some fundamental concepts about algebraic equations. Think of an algebraic equation like a balanced scale. On one side, you have an expression, and on the other side, you have another expression. The equals sign (=) tells us that both sides are perfectly balanced or equal in value. Our goal in solving equations is to figure out what value of the variable (in this case, x) will keep the scale balanced. The variable is like a mystery number we need to uncover.

To keep the equation balanced, whatever operation we perform on one side, we must also perform on the other side. This principle is crucial for solving equations correctly. We can add, subtract, multiply, or divide, but we always have to do the same thing to both sides. This ensures that the equation remains equivalent and the balance is maintained. For instance, if we add a number to the left side, we must add the same number to the right side. This might seem simple, but it's the golden rule of equation solving. Understanding this principle is the first step in mastering algebra.

When solving equations, we often encounter different terms. A term is a single number or variable, or numbers and variables multiplied together. Like terms are terms that contain the same variable raised to the same power. For example, 3x and -5x are like terms, but 3x and 3x² are not. Combining like terms is a vital step in simplifying equations. It involves adding or subtracting the coefficients (the numbers in front of the variables) of like terms. This simplifies the equation and makes it easier to solve.

Another key concept is the distributive property. This property allows us to multiply a single term by each term inside a set of parentheses. For example, a(b + c) is equal to ab + ac. The distributive property is particularly useful when dealing with equations that have parentheses, like the one we're about to solve. It allows us to eliminate the parentheses and simplify the equation into a more manageable form. Remember, mastering these basic concepts is key to confidently tackling more complex algebraic problems.

Step 1: Applying the Distributive Property

Alright, let's dive into our equation: -4(x-7) = 5(-2x-4). The first thing we need to do is get rid of those parentheses. How do we do that? By using the distributive property! Remember, the distributive property allows us to multiply the term outside the parentheses by each term inside. So, on the left side, we'll multiply -4 by both x and -7. On the right side, we'll multiply 5 by both -2x and -4.

Let's start with the left side: -4 multiplied by x is -4x. Then, -4 multiplied by -7 is +28. Remember, a negative times a negative is a positive! So, the left side of the equation becomes -4x + 28. Now, let's tackle the right side: 5 multiplied by -2x is -10x. And 5 multiplied by -4 is -20. So, the right side becomes -10x - 20. Our equation now looks like this: -4x + 28 = -10x - 20. See? We've taken the first big step by distributing those numbers and making the equation look a bit cleaner. This step is crucial because it allows us to combine like terms later on.

The distributive property is like a secret weapon in algebra. It helps us transform complex-looking expressions into simpler ones. By multiplying the term outside the parentheses by each term inside, we effectively eliminate the parentheses and create a more manageable equation. It's not just about multiplying numbers; it's about understanding how mathematical operations interact with each other. For instance, paying attention to signs (positive and negative) is super important. A simple sign error can throw off your entire solution, so always double-check your work!

Once you become comfortable with the distributive property, you'll start seeing it everywhere in algebra problems. It's one of those fundamental skills that you'll use over and over again. So, take your time, practice it, and make sure you understand it thoroughly. And remember, math isn't just about getting the right answer; it's about understanding the process. So, let's move on to the next step and keep solving this equation!

Step 2: Combining Like Terms and Isolating the Variable

Okay, guys, we've distributed the numbers and now our equation looks like this: -4x + 28 = -10x - 20. The next step is to gather all the x terms on one side of the equation and all the constant terms (the numbers without variables) on the other side. This is called isolating the variable, and it's a crucial step in solving for x. To do this, we'll use the addition and subtraction properties of equality, which basically say that we can add or subtract the same thing from both sides of the equation without changing its balance.

Let's start by moving the -10x term from the right side to the left side. To do this, we'll add 10x to both sides of the equation. Remember, whatever we do to one side, we have to do to the other! So, we have: -4x + 10x + 28 = -10x + 10x - 20. Simplifying this gives us 6x + 28 = -20. See how the -10x and +10x on the right side canceled each other out? That's exactly what we wanted!

Now, let's move the constant term, +28, from the left side to the right side. To do this, we'll subtract 28 from both sides: 6x + 28 - 28 = -20 - 28. Simplifying this gives us 6x = -48. We're getting closer! We've managed to get all the x terms on one side and all the constant terms on the other. The equation is much simpler now, and we're just one step away from finding the value of x.

Isolating the variable is like peeling away the layers of an onion. Each time we add, subtract, multiply, or divide, we're getting closer to revealing the value of x. It's a strategic process, and it requires careful attention to detail. Remember to always perform the same operation on both sides of the equation to maintain balance. This might seem tedious, but it's what ensures that our solution is correct. So, let's move on to the final step and solve for x!

Step 3: Solving for x by Division

We're almost there, guys! Our equation is now 6x = -48. We've done a great job of simplifying it, and now we just need to isolate x completely. Right now, x is being multiplied by 6. To undo this multiplication and get x by itself, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 6.

So, we have (6x) / 6 = -48 / 6. On the left side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with just x. On the right side, -48 divided by 6 is -8. Therefore, our solution is x = -8. Hooray! We've solved the equation!

It's always a good idea to double-check our answer to make sure it's correct. To do this, we'll substitute x = -8 back into the original equation: -4(x-7) = 5(-2x-4). Plugging in -8 for x, we get -4(-8-7) = 5(-2(-8)-4). Let's simplify each side: -4(-15) = 5(16-4), which becomes 60 = 5(12), and finally, 60 = 60. The left side equals the right side, so our solution x = -8 is correct!

Solving for x by division is the final flourish in our algebraic dance. It's the moment where we unveil the mystery number and complete the puzzle. Remember, division is the opposite of multiplication, and it's a powerful tool for isolating variables. Always double-check your work by plugging your solution back into the original equation. This not only confirms your answer but also reinforces your understanding of the equation-solving process.

Conclusion: Mastering the Art of Equation Solving

Fantastic job, everyone! We've successfully solved the equation -4(x-7) = 5(-2x-4) and found that x = -8. We've journeyed through the distributive property, combining like terms, and isolating the variable. You've seen how each step builds upon the previous one, leading us to the solution. But more than just finding the answer, we've explored the process of equation solving, which is a fundamental skill in mathematics and beyond.

Solving equations is like learning a new language. At first, it might seem daunting, with all the terms, variables, and operations. But with practice and patience, you'll start to recognize patterns, understand the rules, and develop your own fluency. The key is to break down complex problems into smaller, manageable steps. Remember, we started with a seemingly complicated equation, but by applying the distributive property, combining like terms, and isolating the variable, we transformed it into a simple one-step problem.

The skills you've learned today are not just for solving algebraic equations. They're transferable skills that can help you in many areas of life. Problem-solving, logical reasoning, and attention to detail are valuable assets in any field. So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is vast and fascinating, and there's always something new to discover.

Whether you're tackling a tough math problem, making a decision at work, or navigating a complex situation in your personal life, the ability to break things down, analyze them logically, and find solutions is invaluable. So, pat yourself on the back for conquering this equation, and get ready for the next mathematical adventure! You've got this!